A new construction of p-adic Rankin convolutions in the case of positive slope

TL;DR

This paper extends the construction of p-adic Rankin convolutions to cases where the modular form has positive slope, beyond the ordinary case, using Panchishkin's method.

Contribution

It introduces a new method to construct p-adic Rankin convolutions for modular forms with small positive slope, generalizing previous work limited to ordinary forms.

Findings

Constructed p-adic L-functions for small slope forms

Generalized Hida's construction beyond ordinary forms

Validated the method for specific modular forms

Abstract

Given two newforms $f$ and $g$ of respective weights $k$ and $l$ with $k<l$, Hida constructed a $p$-adic $L$-function interpolating the values of the Rankin convolution of $f$ and $g$ in the critical strip $l \leq s \leq k$. However, this construction works only if $f$ is an ordinary form. Using a method developed by Panchishkin to construct $p$-adic $L$-function associated with modular forms, we generalize this construction to the case where the slope of $f$ is small.